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High Moment Partial Sum Processes of Residuals in ARMA Models and their Applications

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  • Hao Yu

Abstract

. In this article, we study high moment partial sum processes based on residuals of a stationary autoregressive moving average (ARMA) model with known or unknown mean parameter. We show that they can be approximated in probability by the analogous processes which are obtained from the i.i.d. errors of the ARMA model. However, if a unknown mean parameter is used, there will be an additional term that depends on model parameters and a mean estimator. When properly normalized, this additional term will vanish. Thus the processes converge weakly to the same Gaussian processes as if the residuals were i.i.d. Applications to change‐point problems and goodness‐of‐fit are considered, in particular, cumulative sum statistics for testing ARMA model structure changes and the Jarque–Bera omnibus statistic for testing normality of the unobservable error distribution of an ARMA model.

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  • Hao Yu, 2007. "High Moment Partial Sum Processes of Residuals in ARMA Models and their Applications," Journal of Time Series Analysis, Wiley Blackwell, vol. 28(1), pages 72-91, January.
    • Handle: RePEc:bla:jtsera:v:28:y:2007:i:1:p:72-91
    DOI: 10.1111/j.1467-9892.2006.00499.x
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    Cited by:

    1. Sucarrat, Genaro & Grønneberg, Steffen & Escribano, Alvaro, 2016. "Estimation and inference in univariate and multivariate log-GARCH-X models when the conditional density is unknown," Computational Statistics & Data Analysis, Elsevier, vol. 100(C), pages 582-594.
    2. Michael W. Robbins & Colin M. Gallagher & Robert B. Lund, 2016. "A General Regression Changepoint Test for Time Series Data," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 111(514), pages 670-683, April.
    3. Daniel Philps & Artur d'Avila Garcez & Tillman Weyde, 2019. "Making Good on LSTMs' Unfulfilled Promise," Papers 1911.04489, arXiv.org, revised Dec 2019.
    4. Christopher Dienes & Alexander Aue, 2014. "On-Line Monitoring Of Pollution Concentrations With Autoregressive Moving Average Time Series," Journal of Time Series Analysis, Wiley Blackwell, vol. 35(3), pages 239-261, May.
    5. Song, Junmo & Kang, Jiwon, 2018. "Parameter change tests for ARMA–GARCH models," Computational Statistics & Data Analysis, Elsevier, vol. 121(C), pages 41-56.
    6. Raïssi, Hamdi, 2018. "Testing normality for unconditionally heteroscedastic macroeconomic variables," Economic Modelling, Elsevier, vol. 70(C), pages 140-146.
    7. Daniel Philps & Tillman Weyde & Artur d'Avila Garcez & Roy Batchelor, 2018. "Continual Learning Augmented Investment Decisions," Papers 1812.02340, arXiv.org, revised Jan 2019.

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